Peaks and jumps reconstruction with BB-splines scaling functions

We consider a methodology based on BB-splines scaling functions to numerically invert Fourier or Laplace transforms of functions in the space L2(R)L2(R). The original function is approximated by a finite combination of jjth order BB-splines basis functions and we provide analytical expressions for the recovered coefficients. The methodology is particularly well suited when the original function or its derivatives present peaks or jumps due to discontinuities in the domain. We will show in the numerical experiments the robustness and accuracy of the method.